Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

HU, Yong

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2131-2143.

Soit $R$ un anneau local intègre de dimension $2$, normal, excellent et hensélien dans lequel $2$ est inversible. Soient $L$ son corps de fractions et $k$ son corps résiduel. Soit ${\Omega }_R$ l’ensemble des valuations discrètes de rang 1 de $L$ correspondant aux points de codimension 1 des modèles propres réguliers de $SpecR$. On démontre qu’une forme quadratique $q$ sur $L$ satisfait le principe local-global par rapport à ${\Omega }_R$ dans les deux cas suivants : (1) $q$ est de rang 3 ou 4 ; (2) $q$ est de rang $\ge 5$ et $R=A\left[\right[y\left]\right]$, où $A$ est un anneau de valuation discrète complet, avec une condition sur le corps résiduel $k$ qui est satisfaite lorsque $k$ est $C_1$.

Let $R$ be a 2-dimensional normal excellent henselian local domain in which $2$ is invertible and let $L$ and $k$ be its fraction field and residue field respectively. Let ${\Omega }_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $SpecR$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to ${\Omega }_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A\left[\right[y\left]\right]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.

Classification : 11E04, 11E08, 11D88, 14G99
Keywords: 2-dimensional local ring, local-global principle, quadratic forms, complete local domain
Mot clés : anneau local de dimension 2, principe local-global, formes quadratiques, anneau local complet

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